HOT topics, or higher order thinking topics, are lessons or activities used to get students creatively thinking instead of spitting back knowledge. For most adults, higher order thinking seems to come naturally since life is not a series of questions with definite answers, but instead is a set of problems to solve. School, however, tends to be filled with more knowledge based assessments rather than problems that require higher order thinking skills. This can cause issues later in life when students encounter real problems where the solution is not obvious. When teachers plan lessons, very little should involve memorizing facts, but should rather most should require problem solving skills and creative thinking. This allows students to learn the skills necessary to use the resources available in order to find solutions to any problems. Higher order thinking requires three necessary ideas: transfer, critical thinking, and problem solving. In order for students to be able to transfer knowledge, they must understand it enough to take it to another level and apply it to an unknown problem. Critical thinking requires students to actively use their knowledge in order to make helpful decisions. Finally, problem solving involves encountering a problem that does not have an immediately understood solution, but instead requires analyzing information and deciding how best to use that information to reach a solution.
The best way to encourage higher order thinking is to let students know what it is and how to use it to solve problems. The following are strategies for teachers to develop higher order thinking skills in their classes. First, encourage questioning. Students must feel safe in order to take risks and question ideas instead of always wanting to be given the answers. Next, connect concepts. This allows students to relate new ideas to things they already understand. Furthermore, teach students to infer by giving them real world problems for them to solve. Give students tools such as graphic organizers to help them organize their thoughts and ideas into a way to reach a solution. Teach problem solving strategies. This can be done by giving students many different types of problems and showing them different strategies that can be used. Encourage creative thinking, by giving students problems or brain teasers that require them to come up with strategies they have never used before. Have students elaborate on their answers by requiring them to show understanding by justifying their answers. Let students know that the answer itself is the least important part of the solution, how and why they got there is a much bigger part of understanding.
Resources:
Brookhart, Susan M. "How to Assess Higher-Order Thinking Skills in Your Classroom." Introduction. ASCD, 2010. Web. 27 Nov. 2016.
Cox, Janelle. "Teaching Strategies That Enhance Higher-Order Thinking." TeachHUB. N.p., n.d. Web. 27 Nov. 2016.
Thomas, Alice, and Glenda Thorne. "How To Increase Higher Level Thinking | Center for Development and Learning." Center for Development and Learning RSS2. N.p., 7 Dec. 2009. Web. 27 Nov. 2016.
List of HOT Lessons:
1. Puzzles and Brainteasers
I give these puzzles to my students as a warm up as soon as class starts. I have found many of the warm ups that I use at this site. I tell my students that they will receive one point just for coming to class on time and working on the warm up. They will receive one point of extra credit if they get the correct answer. This allows the students to work on the problem creatively without fear of getting the wrong answer. I try to choose problems that target specific problem solving strategies, but that do not involve a known, mathematical algorithm so that students must think about the best strategy to choose. Finally, when everyone has answered, I ask the students that got the correct answer what strategy they used. Many times, there are several different strategies, and often, students come up with solutions that even I had not thought of. We also go over what incorrect thinking was used when students get wrong answers. This helps students to avoid common mistakes in mathematical thinking.
This is a webquest that focuses on area, ratios, and volume and also incorporates information about animals and their habitats. The goal of the activity is for students to design a temporary holding tank for green sea turtles and hammerhead sharks. Students must use their understanding of the geometry topics as well as research and use the information they find about the animals they are designing a habitat for. The reason that this activity is a HOT activity is because students must use creativity and problem solving skills to design something that can be used in real life. Furthermore, after the drawing and 3D image are created students must think about the process they used to design the pool and extend their ideas into other areas.
Kaiser, Jared. "Problem at the Aquarium." Problem at the Aquarium: Introduction. N.p., n.d. Web. 28 Nov. 2016. http://questgarden.com/191/93/7/161108110828/index.htm
Grade Level: 9-12 Content Area: Mathematics/Geometry
3. Ask The Sun
This is another webquest. This one uses the idea that people can use indirect measurement (in this case shadows) to figure out the heights of things that are impractical to measure directly. Students must work together and use what they know about similar triangles and proportions to solve for the heights of the objects they choose. This is again a higher order thinking activity because it is a real life situation that requires students to use skills that they have learned to solve a problem. After students finish the mathematical part of the activity, they must create a presentation to show how they accomplished their task. This allows students to be creative and add a little flair to the project. Finally, students are asked to take the ideas further and research and learn about the pyramid of Giza. They then are asked to apply what they did previously to this new situation.
In this activity on Desmos, students watch a video of someone filling a small circle with pennies. Then someone begins to fill a larger circle and the students are asked to guess how many pennines will fill the circle. Next students actually collect data by dragging online pennies into different sized circles. Finally, students try out different models for their data to see what fits it best. The goals of this activity are for students to learn the strategy of using easier models to predict what will happen in more complex models and also to understand the difference between linear, quadratic, and exponential models. This is a HOT topic because again students are using real life data and questions to help them understand mathematical models and why they are useful. Students must reason through, make conjectures, and go back and start again if their ideas do not work the first time.
Kane, Dylan. "Penny Circle by Desmos." Penny Circle by Desmos. N.p., n.d. Web. 28 Nov. 2016. https://teacher.desmos.com/pennycircle
Grade Level: 9-11 Content Area: Mathematics/Algebra
5. Find the Soap
In this activity, students are taught what a locus is and then are taught about different types of loci by using questioning, imagination, prediction, and justification. In each case, students are asked to create what they think the specific locus would look like. Then they see what it actually looks like and are asked to make a prediction. Then they test their prediction and justify why they were correct or explain what was wrong. This is a HOT activity because students must use their imaginations to create things before they even know what they are. Then they must use their understanding to make predictions and test those hypotheses. Finally, in the challenges, they must apply their understanding to new, more complicated situations.
This activity is designed to help students learn vocabulary without memorization. The idea of the game is to have students pair up and play a "guess who" type of game with different related angles. One student picks a picture and answers questions about it. The other student asks questions trying to guess which picture the first student chose. This is definitely a HOT activity because students must think of good questions to ask, why they are helpful questions, and must have a good understanding of the different angle relationships in order to know what questions to ask and to figure the picture out from the answers to their questions. Instead of memorizing definitions and looking at pictures, students must use critical thinking skills and imagination and show they they truly understand the different types of angle pairs.
In this activity, students build triangles choosing 3 given parts and try to see if the triangles will always be congruent or not. Students learn the difference between SSS, SAS, SSA, ASA, AAS, and AAA and will be able to see if it is possible to make triangles that are not congruent. Because sometimes students will make congruent triangles with SSA and AAA, even those these do not prove triangle congruence, students must use critical thinking to decide if one time is enough to make a conjecture or if they should try more than once and/or share results with peers. Finally, students are able to make conjectures for themselves about what is need to show know that triangles are congruent. Students can further their understanding with the proof activity that goes along with this.
In this activity, students measure the circumference of a lollipop and calculate the radius. Next, they suck on the lollipop for 120 seconds and take a new measurement. They repeat this process until they have a table full of data. After they fill the table, students are asked to decide what type of model is best for their plotted points. Then they use their calculator to make a regression line and use the equation to explain what the rate of change of the radius of their sucker is. Finally, they are to use this rate of change and the volume of a sphere to come up with the rate of change of the volume. This is a HOT activity because students apply their knowledge of rates of change to a real life situation and have to evaluate their steps and models and explain what they mean.
"Teaching and Assessing Calculus." AcademicMerit. College Board, n.d. Web. 28 Nov. 2016.
In this activity, students are asked to take a Road Runner cartoon and evaluate different situations that occur using calculus. Students must think about what is going on in the cartoon and calculate maximums, minimums, points of inflection, the depth of the water under the mountain, and the distance between the road runner and the coyote. This is a HOT activity because students are applying strategies and ideas that they have learned in calculus to a situation. They have to think about how all the things they have learned about using the first and second derivative apply to this cartoon. They can even watch the cartoon on this site below to see their calculus in action.
This is a lesson I designed for my students that I used when presenting my flipped classroom at Interface last year. My students use The Geometer's Sketchpad but since I wanted to share, I also made the quadrilaterals in Geogebra. In this lesson, students explore the quadrilaterals and discover properties about each special quadrilateral. Using sketchpad or geogebra, students can measure side lengths, angle measures, diagonal lengths, pieces of diagonals, etc. and make conjectures about things they think are true about those quadrilaterals. They can move the quadrilaterals around and change the measures to see if their conjectures stay true. When they think they know the properties, I ask them to fill out the chart of quadrilateral properties (see below). Next, I ask them to visit the illuminations link below and play with the different kinds of quadrilaterals formed with different conditions for the diagonals. Then I ask them to answer the questions on the rhombus and rectangle investigation sheet (see below) telling me whether each statement is true or false and if false, to draw a counterexample. From that activity, the students should be able to come up with the conditions needed to tell if a quadrilateral is a rectangle, rhombus, or square. Finally, I tie it to the real world by asking the students to explain when testing for these conditions would be useful in real life situations.
HOT (Higher Order Thinking) Topics
Table of Contents
Summary:
HOT topics, or higher order thinking topics, are lessons or activities used to get students creatively thinking instead of spitting back knowledge. For most adults, higher order thinking seems to come naturally since life is not a series of questions with definite answers, but instead is a set of problems to solve. School, however, tends to be filled with more knowledge based assessments rather than problems that require higher order thinking skills. This can cause issues later in life when students encounter real problems where the solution is not obvious. When teachers plan lessons, very little should involve memorizing facts, but should rather most should require problem solving skills and creative thinking. This allows students to learn the skills necessary to use the resources available in order to find solutions to any problems. Higher order thinking requires three necessary ideas: transfer, critical thinking, and problem solving. In order for students to be able to transfer knowledge, they must understand it enough to take it to another level and apply it to an unknown problem. Critical thinking requires students to actively use their knowledge in order to make helpful decisions. Finally, problem solving involves encountering a problem that does not have an immediately understood solution, but instead requires analyzing information and deciding how best to use that information to reach a solution.The best way to encourage higher order thinking is to let students know what it is and how to use it to solve problems. The following are strategies for teachers to develop higher order thinking skills in their classes. First, encourage questioning. Students must feel safe in order to take risks and question ideas instead of always wanting to be given the answers. Next, connect concepts. This allows students to relate new ideas to things they already understand. Furthermore, teach students to infer by giving them real world problems for them to solve. Give students tools such as graphic organizers to help them organize their thoughts and ideas into a way to reach a solution. Teach problem solving strategies. This can be done by giving students many different types of problems and showing them different strategies that can be used. Encourage creative thinking, by giving students problems or brain teasers that require them to come up with strategies they have never used before. Have students elaborate on their answers by requiring them to show understanding by justifying their answers. Let students know that the answer itself is the least important part of the solution, how and why they got there is a much bigger part of understanding.
Resources:
Brookhart, Susan M. "How to Assess Higher-Order Thinking Skills in Your Classroom." Introduction. ASCD, 2010. Web. 27 Nov. 2016.
Cox, Janelle. "Teaching Strategies That Enhance Higher-Order Thinking." TeachHUB. N.p., n.d. Web. 27 Nov. 2016.
Thomas, Alice, and Glenda Thorne. "How To Increase Higher Level Thinking | Center for Development and Learning." Center for Development and Learning RSS2. N.p., 7 Dec. 2009. Web. 27 Nov. 2016.
List of HOT Lessons:
1. Puzzles and Brainteasers
I give these puzzles to my students as a warm up as soon as class starts. I have found many of the warm ups that I use at this site. I tell my students that they will receive one point just for coming to class on time and working on the warm up. They will receive one point of extra credit if they get the correct answer. This allows the students to work on the problem creatively without fear of getting the wrong answer. I try to choose problems that target specific problem solving strategies, but that do not involve a known, mathematical algorithm so that students must think about the best strategy to choose. Finally, when everyone has answered, I ask the students that got the correct answer what strategy they used. Many times, there are several different strategies, and often, students come up with solutions that even I had not thought of. We also go over what incorrect thinking was used when students get wrong answers. This helps students to avoid common mistakes in mathematical thinking.
"BrainBashers : Puzzles and Brain Teasers." BrainBashers : Puzzles and Brain Teasers. N.p., n.d. Web. 27 Nov. 2016.
http://www.brainbashers.com/puzzles.asp
Grade Level: 9-12 Content Area: Mathematics
2. Problem at the Aquarium
This is a webquest that focuses on area, ratios, and volume and also incorporates information about animals and their habitats. The goal of the activity is for students to design a temporary holding tank for green sea turtles and hammerhead sharks. Students must use their understanding of the geometry topics as well as research and use the information they find about the animals they are designing a habitat for. The reason that this activity is a HOT activity is because students must use creativity and problem solving skills to design something that can be used in real life. Furthermore, after the drawing and 3D image are created students must think about the process they used to design the pool and extend their ideas into other areas.
Kaiser, Jared. "Problem at the Aquarium." Problem at the Aquarium: Introduction. N.p., n.d. Web. 28 Nov. 2016.
http://questgarden.com/191/93/7/161108110828/index.htm
Grade Level: 9-12 Content Area: Mathematics/Geometry
3. Ask The Sun
This is another webquest. This one uses the idea that people can use indirect measurement (in this case shadows) to figure out the heights of things that are impractical to measure directly. Students must work together and use what they know about similar triangles and proportions to solve for the heights of the objects they choose. This is again a higher order thinking activity because it is a real life situation that requires students to use skills that they have learned to solve a problem. After students finish the mathematical part of the activity, they must create a presentation to show how they accomplished their task. This allows students to be creative and add a little flair to the project. Finally, students are asked to take the ideas further and research and learn about the pyramid of Giza. They then are asked to apply what they did previously to this new situation.
Cordova, Daniel. "Ask The Sun." Ask The Sun: Introduction. N.p., n.d. Web. 28 Nov. 2016.
http://questgarden.com/189/93/4/160625012744/index.htm
Grade Level: 6-10 Content Area: Mathematics/Geometry
4. Penny Circle
In this activity on Desmos, students watch a video of someone filling a small circle with pennies. Then someone begins to fill a larger circle and the students are asked to guess how many pennines will fill the circle. Next students actually collect data by dragging online pennies into different sized circles. Finally, students try out different models for their data to see what fits it best. The goals of this activity are for students to learn the strategy of using easier models to predict what will happen in more complex models and also to understand the difference between linear, quadratic, and exponential models. This is a HOT topic because again students are using real life data and questions to help them understand mathematical models and why they are useful. Students must reason through, make conjectures, and go back and start again if their ideas do not work the first time.
Kane, Dylan. "Penny Circle by Desmos." Penny Circle by Desmos. N.p., n.d. Web. 28 Nov. 2016.
https://teacher.desmos.com/pennycircle
Grade Level: 9-11 Content Area: Mathematics/Algebra
5. Find the Soap
In this activity, students are taught what a locus is and then are taught about different types of loci by using questioning, imagination, prediction, and justification. In each case, students are asked to create what they think the specific locus would look like. Then they see what it actually looks like and are asked to make a prediction. Then they test their prediction and justify why they were correct or explain what was wrong. This is a HOT activity because students must use their imaginations to create things before they even know what they are. Then they must use their understanding to make predictions and test those hypotheses. Finally, in the challenges, they must apply their understanding to new, more complicated situations.
Jorgens, Paul. "Find the SOAP." Desmos Classroom Activities. N.p., n.d. Web. 28 Nov. 2016.
https://teacher.desmos.com/activitybuilder/custom/573a63dc1540836b38b6254a#
Grade Level: 9-12 Content Area: Mathematics
6. Polygraph: Angle Relationships
This activity is designed to help students learn vocabulary without memorization. The idea of the game is to have students pair up and play a "guess who" type of game with different related angles. One student picks a picture and answers questions about it. The other student asks questions trying to guess which picture the first student chose. This is definitely a HOT activity because students must think of good questions to ask, why they are helpful questions, and must have a good understanding of the different angle relationships in order to know what questions to ask and to figure the picture out from the answers to their questions. Instead of memorizing definitions and looking at pictures, students must use critical thinking skills and imagination and show they they truly understand the different types of angle pairs.
Nur, Laila. "Angle Relationships." Desmos Classroom Activities. N.p., n.d. Web. 28 Nov. 2016.
https://teacher.desmos.com/polygraph/custom/560c53f831e47ee40c824ed0
Grade Level: 8-11 Content Area: Mathematics/Geometry
7. Congruence Theorems
In this activity, students build triangles choosing 3 given parts and try to see if the triangles will always be congruent or not. Students learn the difference between SSS, SAS, SSA, ASA, AAS, and AAA and will be able to see if it is possible to make triangles that are not congruent. Because sometimes students will make congruent triangles with SSA and AAA, even those these do not prove triangle congruence, students must use critical thinking to decide if one time is enough to make a conjecture or if they should try more than once and/or share results with peers. Finally, students are able to make conjectures for themselves about what is need to show know that triangles are congruent. Students can further their understanding with the proof activity that goes along with this.
Illuminations.nctm.org. NCTM, n.d. Web. 28 Nov. 2016.
http://illuminations.nctm.org/activity.aspx?id=3504
Grade Level: 9-12 Content: Mathematics/Geometry
8. The Sucker Project
In this activity, students measure the circumference of a lollipop and calculate the radius. Next, they suck on the lollipop for 120 seconds and take a new measurement. They repeat this process until they have a table full of data. After they fill the table, students are asked to decide what type of model is best for their plotted points. Then they use their calculator to make a regression line and use the equation to explain what the rate of change of the radius of their sucker is. Finally, they are to use this rate of change and the volume of a sphere to come up with the rate of change of the volume. This is a HOT activity because students apply their knowledge of rates of change to a real life situation and have to evaluate their steps and models and explain what they mean.
"Teaching and Assessing Calculus." AcademicMerit. College Board, n.d. Web. 28 Nov. 2016.
Grade Level: 11-12 Content: Mathematics/Calculus
9. Max/Min With Wile E Coyote
In this activity, students are asked to take a Road Runner cartoon and evaluate different situations that occur using calculus. Students must think about what is going on in the cartoon and calculate maximums, minimums, points of inflection, the depth of the water under the mountain, and the distance between the road runner and the coyote. This is a HOT activity because students are applying strategies and ideas that they have learned in calculus to a situation. They have to think about how all the things they have learned about using the first and second derivative apply to this cartoon. They can even watch the cartoon on this site below to see their calculus in action.
Roberts, Donna. "Math and the Movies Resource List." Math and the Movies Resource List. N.p., n.d. Web. 28 Nov. 2016.
http://mathbits.com/MathBits/MathMovies/ResourceListTwo.html
Grade Level: 11-12 Content Area: Mathematics/Calculus
10. Exploring Quadrilaterals
This is a lesson I designed for my students that I used when presenting my flipped classroom at Interface last year. My students use The Geometer's Sketchpad but since I wanted to share, I also made the quadrilaterals in Geogebra. In this lesson, students explore the quadrilaterals and discover properties about each special quadrilateral. Using sketchpad or geogebra, students can measure side lengths, angle measures, diagonal lengths, pieces of diagonals, etc. and make conjectures about things they think are true about those quadrilaterals. They can move the quadrilaterals around and change the measures to see if their conjectures stay true. When they think they know the properties, I ask them to fill out the chart of quadrilateral properties (see below). Next, I ask them to visit the illuminations link below and play with the different kinds of quadrilaterals formed with different conditions for the diagonals. Then I ask them to answer the questions on the rhombus and rectangle investigation sheet (see below) telling me whether each statement is true or false and if false, to draw a counterexample. From that activity, the students should be able to come up with the conditions needed to tell if a quadrilateral is a rectangle, rhombus, or square. Finally, I tie it to the real world by asking the students to explain when testing for these conditions would be useful in real life situations.
https://www.geogebra.org/m/kmqjJYmM
Illuminations.nctm.org. NCTM, n.d. Web. 30 Nov. 2016.
http://illuminations.nctm.org/Activity.aspx?id=3578
Grade Level: 9-12 Content Area: Mathematics/Geometry